at a time t+δt , where δt is a small non-zero finite time that may be any multiple of the electron's intrinsic clock-time or chronon – see [%??]. Vn ≡ V(tn+δt) = V+ n.
FINITE MECHANICS UET7C.341 – a draft of a section of the paper: “QUANTUM ELECTRON MECHANICS (QEM)” © H. J. Spencer (December, 2017)
ABSTRACT The author has been developing a new approach to mechanics (for 10 years) that replaces the orthodox techniques that were introduced by Isaac Newton in his Principia (1687) that formed the foundations of Classical Mechanics (CM). Newton’s approach incorporated a new metaphysics based on the revised ideas of Greek Atomists plus new mathematical methods incorporating the controversial concept of infinitesimals. This reflected the metaphysical assumption that particles were subject to continuous interactions, we call Forces. This assumption is referred to here as the Continuum Hypothesis, which cannot be tested at the laboratory scale but still resulted in the abstractions known as instantaneous velocity and Newton’s major physical innovation: instantaneous particle momentum (or ‘Quantity of Motion’). Actually, some scholars have pointed out that Newton first approached these problems with the idea of discrete interactions occurring discretely in time (called Impulses) but jumped to the limit of zero time separations, so as to solve the apparent continuous motions of the planets in his radical gravity model. Newton’s ideas were found not to work at the atomic scale of electrons, where discrete (quantum) constraints were needed. In 1913, Bohr achieved revolutionary success by simply quantizing the atom’s angular momentum around a circular orbit, while retaining the rest of CM. Little known (or studied) today, is the fact that Sommerfeld achieved spectacular accuracy (calculating the observed ‘Fine Structure’ of the hydrogen atom) by extending Bohr’s orbits to elliptical shapes, as Kepler had observed and Newton had calculated in his two-body planetary model. Sommerfeld’s 2D relativistic results were certainly much superior to the later Schrödinger’s (non-relativistic) Wave Mechanics and theorists had to await Dirac’s unique mathematical Theory of the Electron (applied as a 3D spherical CM approach) before Sommerfeld’s were matched. Even so, (mathematical) theoretical-physicists preferred the elegance of the New Quantum Mechanics (QM) so little attention has been given to Sommerfeld’s ideas in the last 100 years. Even Quantum Field Theory (QFT) stills clings to continuum mathematics. The new QEM approach adopts some of Newton’s original ideas but rejects the Continuum Hypothesis so it is constructed around discrete mathematics and avoids all continuous ideas, like the Calculus. This radical approach is proving successful as will be shown when the full paper is released but the (aging) author believes there was some value in pre-releasing some of these ideas for younger physicists. As a result, this is one of the core sections, illustrating that discrete physics can still exhibit one of Newton’s major results: the centrality of the idea of Conserved Quantities extends across ALL scales of material reality. Hopefully, we will be able to complete this research programme with another 4 papers replacing The Standard Model with a Universal Electron Theory (UET).
3.4.1 CONSERVED QUANTITIES The present theory is based on Newton’s views of invariant space and time; with particles moving through a passive space and inter-particle interactions occurring at only certain points in time. This traditional model rejects all views of both these fundamental ‘background’ concepts either mixing (through a change in an observer’s motion) or becoming distorted due to large quantities of matter. Newton’s mass particles are here seen as point electrons subject to mutual pair-wise interactions that manifest themselves as finite impulses at each participating electron. This theory is relational (or ‘relative’) because the interaction is only based on their relative spatial and temporal separations while the consequence (effect) of the impulse only depends on their relative positions and velocities. This research programme’s views on the fundamental nature of Nature were first introduced in the previous paper [UET6] and will be elaborated here.
3.4.1.1 PAIRWISE INTERACTIONS The following analysis is constructed around the present theory’s metaphysical hypothesis [UET4] of the EM interaction between pairs of electrons. The theory extends (or reinterprets) Newton’s mechanics of a generalized particle, subject to ‘external’ forces (continuous interactions), to occasional interactions (attractions or repulsions) between pairs of electrons at a set of times {tn} as defined in an arbitrary inertial Euclidian Reference Frame (relative to some arbitrary Origin with a Master-Clock, defining the uniform passage of time). Thus, in the following scenario, we can imagine a single electron (the ‘target’) with a universal and invariant intrinsic (inertial) mass m moving freely through 3D space, with a prior velocity u just before the specific time tn, when it is located at the spatial location [X(tn)≡Xn] relative to the Origin. At this instant, the electron becomes involved in an interaction [UET5] with another (remote) electron and receives an impulse [∆I(tn) ≡ ∆In] that causes it to move off with a velocity Vn. This velocity could be subsequently measured (confirmed) by an observation at any time t′ [tn < t′ < tn+1]; implying then that the observation was the very next interaction with the target electron (this is a major distinction with Newton’s method of infinitesimals). We shall now show that in this discrete scenario, the electron maintains several mechanical quantities as constants of the motion during the time interval ∆tn ≡ tn+1 − tn. In the following, we simply assume Newton’s original (particle) Laws of Motion still apply at the scale of the electron; as discrete times are seen as sufficient to provide a comprehensive new form of Quantum (Electron) Mechanics.
3.4.1.1.1 LINEAR MOMENTUM The heart of Finite Mechanics is Newton’s concept of particle momentum, designated P(t) for every moment t when the electron in moving with constant velocity V (Newton’s First Law of Motion), that is established after the last interaction or at a time t+δt , where δt is a small non-zero finite time that may be any multiple of the electron’s intrinsic clock-time or chronon – see [%??]. Vn ≡ V(tn+δt) = V+n Momentum at time t, P(t) ≡ m Vn {tn ≤ t < tn+1} Velocity (at any later time t) is “distance per unit time”: V(t) ≡ {X(t) – X(tn)} / (t – tn) {tn < t < tn+1 } ∴ X(t) = Xn + (t – tn) Vn ∴ Xn+1 = X(tn+1) = Xn + (tn+1 – tn) Vn ∴ Vn = ∆Xn / ∆tn for all finite (non zero) ∆tn . ∴ V(t) = Vn
∴ m V(t) = m Vn ∴ P(t) = Pn
{tn ≤ t < tn+1} ∴ Linear-Momentum conserved between interactions.
Newton’s Second Law, with Impulse not Force, [see UET3] “Impulse changes an electron’s future momentum, at time tn”. ∴ ∆I(tn) = P+n − P−n = m{V(tn+δt) −V(tn−δt)} NB If no other interactions then: V(tn+1−δt) = V(tn+δt) or V−n+1 = V+n . ∴ ∆I(tn) = m (V+n − V−n-1) = m ∆Vn-1 = (P+n − P−n-1) = ∆Pn-1 ≡ ∆In-1 NB subscript difference: ∴ ∆In => ∆Pn 3.4.1.1.2 KINETIC ENERGY (KE) Kinetic energy K, at any non-interacting time t, is defined as the following scalar product: K(t) ≡ ½ V(t) • P(t) {All t ≠ tn}. Thus, for all times between interaction events: V(t) = V+n ∴ K(t) = Kn {tn ≤ t < tn+1} ∴ ∆K(tn+1) = K
+
n+1
−K
−
n+1
∴ Kn ≡ K(tn+δt) = K+n = ½ m (Vn)2 = (Pn)2 /2m
∴ Kinetic Energy is conserved between interactions.
= Kn+1 − Kn = ∆Kn = ½ m {(Vn+1)2 − (Vn)2} = ½ m (Vn+1 + Vn) • Vn = < Vn > • ∆In+1
∴ ◊Kn = K+n − K−n = ½ m { (V+n)2 − (V−n)2} = < Vn > • ∆In+1 3.4.1.1.3 TEMPORAL VARIABILITY The local (relative) state of the electron (but not its send/receipt mode that only affects when it occurs, not how) will determine how an electron responds, as it participates in an interaction. If the impulse is received [with/opposite] the direction of travel at that instant, then the electron’s velocity (and KE) will [increase/decrease]. If the impulse is orthogonal to the direction of travel, only the direction will be changed [away/towards] the interaction partner based on the partner’s electric charge [same/difference]. All mechanical values remain unchanged between interactions, including measurements.
3.4.1.1.4 LINEAR ACTION Linear-action A, AT any non-interacting time t, is defined as the following scalar product: A(t) ≡ X(t) • P(t) {All t ≠ tn}. In reality, only interaction-event times (at the ‘target’ electron) are significant, i.e. the discrete values {tn} ∴ A(tn) ≡ A n . ∴ A(t) = m {Xn + (t – tn) Vn } • V(t) = m {Xn + (t – tn) Vn } • Vn = m Xn • Vn + m (t – tn) Vn • Vn = A(tn) + 2 (t – tn) Kn ∴ A(t) = An + 2 (t – tn) Kn
{So, Action grows linearly with time [like location] for tn ≤ t < tn+1}.
Defining Dynamic Action, as the change in action AT ONE interaction: ADn ≡ A+n − A−n = A(tn+δt) − A(tn−δt) ≡ ◊An ∴ ◊An = ◊{Xn • Pn} = {X+n • P+n − X−n • P−n} = Xn • (P+n − P−n) = Xn • ∆Pn
∴ ADn = Xn • ∆In+1
Defining Kinetic Action, as the action (between two interactions) : AKn ≡ A+n+1 − A−n = A(tn+1−δt ) − A(tn+δt) ≡ ∆An . ∴ AKn = A−n+1 − A+n = {An + 2 (tn+1 – tn) Kn} – {An + 2 (tn+δt – tn) Kn} = 2 (∆tn – δt) Kn
∴ AKn = 2Kn ∆tn = ∆Xn • Pn
Since measurements are interaction-events, then only multiples of the differences are significant: ∆An ≡ An+1 – An . The smallest difference in time that measurements can achieve are called here δt, which is very small but non-zero; Newton → 0. Defining the Total-Linear Action across one interaction interval (∆tn) and before the next one (‘Interval-Action’) i.e. from just before an interaction AT time, t = t−n = (tn − δt) and up to any other time t' (after tn) [i.e. AT time t' = (tn+1−∆tn) ] and before the next interaction at tn+1: ∆An ≡ A(tn+1−δt) − A(tn − δt) = A−n+1 − A−n ∴ ∆An = A(tn+1−δt) − A(tn − δt) ± A(tn + δt) = {A(tn+1−δt) − A(tn + δt)} + {A(tn + δt) − A(tn + δt)} , ??
∴ A(t) = X(tn) ∴ ∆An = AKn + ADn = (◊ + ∆) An = ∆{A(tn)} = A−n+1 − A−n = ∆A−n = ∆An = 2Kn ∆tn + Xn • ∆In+1 .
3.4.1.1.5 ROTATIONAL ACTION (ANGULAR MOMENTUM) The classical definition of the Angular-Momentum of a point particle (at time t) can be seen to be related to Linear Action by replacing the simple Scalar-Product, of the two instantaneous vectors for (relative) Location, X(t) and Momentum P(t), with the analogous Vector-Product. Dimensionally, both quantities only involve the Maxwellian dimensions [M][L2][T-1], which are the dimensions of Action; thus the justification for viewing Angular Momentum, L(t) as Rotational Action, AR(t). We introduce the Swedish Å (with its mnemonic Volle symbol to remind us the link between circles and rotation), Å(t). Angular Momentum is the vector product version of Linear (Kinetic) Action, so: L(t) = AR(t) = Å ≡ X(t) ∧ P(t) {All t ≠ tn}. ∴ L(t) = L(t+n) = L(tn + δt) = X(tn + δt) ∧ P(tn + δt) = m {X+n + [(tn + δt) – tn] Vn } ∧ V+n = m X+n ∧ V+n = Ån . ∴ L(t) = L+n = Ån {tn < t ≤ tn+1}. Thus, Angular-Momentum is identified (like Velocity, Momentum and KE) by its last interaction time (tn). ∴ Ln+1 = L(tn+1) = m X+n+1 ∧ V+n+1 = X+n+1 ∧ Pn+1 = X+n+1 ∧ {P−n+1 + ∆In+1} = X+n+1 ∧ {P+n + ∆In} = Xn+1 ∧ {m Vn + ∆In} = (Xn + ∆Xn) ∧ {m Vn + ∆In+1} = (Xn + ∆tn Vn) ∧ {m Vn + ∆In+1} = m Xn ∧ Vn + {Xn + ∆Xn} ∧ ∆In = Xn ∧ Pn + Xn+1 ∧ ∆In+1 = Ln + Xn+1 ∧ ∆In = Ln + ∆Ln ∆Ln = ∆{X(t) ∧ Pn } = Xn ∧ ∆Pn ≡ ∆Jn = Xn+1 ∧ ∆In
‘Twist’ (Digital Rotational Impulse)
∴ Angular Momentum is conserved between interactions. The analogy between linear/rotational momentum can be extended by explicitly introducing “Dynamic Rotational Action”. Dynamic Rotational Action is the change in rotational action AT ONE interaction: ÅDn ≡ Å+n − Å−n = Å(tn+δt) − Å(tn−δt) ≡ ◊Å n = Xn • ∆In+1 Defining Kinetic-Rotational Action, as the action (between two interactions) : ÅKn ≡ Å+n+1 − Å−n = Å(tn+1−δt) − Å(tn+δt) ≡ ∆Ån = ∆Xn ∧ Pn = 0 (as Pn = m Vn = m ∆Xn / ∆tn ) Defining the Total-Rotational Action across one interaction interval (∆tn) and before the next one (‘Interval-Action’) i.e. from just before an interaction AT time, t = t−n = (tn − δt) and up to any other time t' (after tn) [i.e. AT time t' = (tn+1−∆tn) ] and before the next interaction at tn+1: ∆Ån ≡ Å(tn+1−δt) − Å(tn − δt) = Å−n+1 − Å−n = ÅKn + ÅDn = L+n + ∆Jn
3.4.1.1.6 VECTOR ACTION Since electron interactions can change its linear and rotational motion (in 3D space), which is reflected algebraically in vector algebra by the scalar-product (resulting in a scalar – non-vector) and the vector-product, resulting in another 3D vector. This is one of the reasons for inventing Natural Vectors (UET1) to combine both components as 4-part objects, with the scalar part being imaginary, reflecting the ontological distinction between space and time (time is NOT simply a 1D type of space: contra Einstein et al.) In the present theory, these two distinctive features of reality are mapped into a 4D Natural Vector X with its finite (segment) difference ∆Xn over a finite difference, ∆tn giving a finite velocity Vn (∆Xn /∆tn) and 4D Natural Vector momentum Pn (mVn) while their simple algebraic product generates both scalar and vector results. The 4D extension of the scalar (linear) action concept has been introduced here called Activity R (Xn Pn). Eventually, this will all be integrated into the most useful dynamic quantity (a 4D Natural Vector), its Activity R characterizing an electron throughout its existence across all of space and time; with combined Linear and Rotational Momentum vectors. We could even define a new space vector A(t), at the location X(t) of an electron (at any time t), with magnitude of Linear-Action A(t) and a direction parallel to the velocity V(t). This is the spatial component of the 4D Activity Natural Vector A(t) . A(t) = A(t) Ê(t)
where V(t) = V(t) Ê(t) .
However, our new quaternion-based Natural Vectors (UET1) eliminate this ancient obsession with space alone and provide a natural 4D (relativistic) basis of all of physics as we have shown in several of the earlier papers in this radical programme (referred to always by their abbreviated titles and file names: ‘UETn ‘ with {n = 1,2, ..., 6}. Readers who are interested can freely find all of the earlier papers on this web-site.
